1.
There exist some generalizations of the Steiner formula that hold true for every closed set, see [1]}. The present form is more restrictive.
2.
If \(S\)  is a convex body, then (REF ) will take the form : for all \(\epsilon >0\)
\(\sigma _2(S^{+\epsilon })=\sigma _2(S)+\epsilon L_1(S)+\pi \epsilon ^2 L_0(S).\) 
\(L_1(S)\)  is just the perimeter \(\sigma _1(S)\)  and \(L_0(S)\)  is the Euler characteristic of \(S\)  which is equal to 1.
If, in addition, the number of irregular points of \(S\)  (points where the support cone is not a half space) is finite, then on the basis of the result of Adler and Taylor, (REF ) follows. Thus a convex body with a finite number of irregular points satisfies the SFH property.
3.
If \(S\)  has a positive reach in the sense that there exists a positive constant \(r\)  such that for all \(t\in S^{+r}\) , \(t\)  has only one projection on \(S\) , then (REF ) is true for all \(\epsilon < r\)  (see [2]}, [3]}). Moreover, if \(S\)  is a set with a piecewise-\(\mathcal {C}^2\)  boundary in the sense of Definition REF  and satisfies \(\kappa (S)<\infty \)  where \(\kappa (S)\)  is defined in (REF ), then (REF ) still holds true (see Appendix).
4.
In the most general cases, the constant \(L_1(S)\)  is the outer Minkowski content of \(S\)  (\(\textnormal {OMC}(S)\) ), which is defined, when it exists, by:
\(\sigma _2(S^{+\epsilon })=\sigma _2(S)+\epsilon \textnormal {OMC}(S)+o(\epsilon ).\) 
For more details, see [2]}. It corresponds to the definition of the perimeter of a curve in convex geometry. It can differ from the length of the boundary of \(S\) , for example in the case of "the square with whiskers" (see Figure REF ).
<FIGURE>In this case, the length of the boundary is equal to the perimeter of the square plus the length of the whiskers, while \(\textnormal {OMC}(S)\)  is equal to the perimeter of the square plus two times the length of the whiskers. In addition it should be noticed that \(L_0(S)\)  is not always equal to the Euler characteristic (see Subsection 3.4).

